In this blog post I want to discuss both the Associative and Commutative Properties of Addition and Multiplication and how student's learn to intentionally use these properties in Gattegno's Textbook 2.

What we mean by the Associative Property is that it doesn't matter how we group addends in addition or factors in multiplication. Back in chapter 3 of Textbook 1, we learned that we can use brackets to show what we did first. We learned that for addition and multiplication it doesn't matter where we put the brackets.

By Commutative we mean that we can shift addends and factors around and it doesn't change anything when calculating.

Associative & Commutative Properties and Towers

In textbook 1, students were introduced to both rectangles and crosses. Some of you may have been tempted to skip building the crosses. Since crosses are extended to towers and are the basis for how students study multiplication, we want to pay attention to how they work and how the Associative and Commutative Properties come into play.

You'll notice that in both Textbook 1 and Textbook 2 Gattegno places a lot of focus on finding the factors of a number. Just like in addition, we can move the factors around (Commutative Property) and group them however we want (Associative Property).

For each multiplication problem, we can set up towers for all of the prime factors and by using both the Associative and Commutative Properties we can find all possible factors and their relationships to both division, fractions and exponents.

I'm going to discuss halving and doubling in depth in a later blog post, but for right now, we can see that every time we multiply by two we are doubling the starting number, each time we remove a 2 we are halving the number. If we remove the three we are dividing by three. Students should be given ample time to explore these relationships. Creating towers and writing math statement for what they are doing and what they observe lays a strong foundation not just for multiplication but also formal algebra later on.

In addition the the basic exercises found in Textbook 2, I recommend frequently playing the Substitution Game and using constraints such as equations must contain only multiplication, or equations can contain only fractions and multiplication. This will solidify both properties in the mind of the learner and remove a lot of the confusion that comes with simplifying problems in formal algebra and you'll be playing with division by default.

In the next blog post, I'll be discussing factoring as well as halving, doubling, and tripling as ways to build numbers and set up milestones for division.